Minimum distance–unbalancedness of the merged graph of $ C_3 $ and a tree

Zhenhua Su; Zikai Tang; Zhenhua Su; Zikai Tang

doi:10.3934/math.2024818

AIMS Mathematics

2024, Volume 9, Issue 7: 16863-16875. doi: 10.3934/math.2024818

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Research article

Minimum distance–unbalancedness of the merged graph of $ C_3 $ and a tree

Zhenhua Su ¹,
Zikai Tang ^{2
,
,}

1.
School of Mathematics and Computational Sciences, Huaihua University, Huaihua, Hunan 418008, China
2.
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

Received: 06 March 2024 Revised: 24 April 2024 Accepted: 08 May 2024 Published: 14 May 2024
MSC : 05C05, 05C09, 05C92

For a graph $ G $, let $ n_G(u, v) $ be the number of vertices of $ G $ that are strictly closer to $ u $ than to $ v $. The distance–unbalancedness index $ {\rm uB}(G) $ is defined as the sum of $ |n_G(u, v)-n_G(v, u)| $ over all unordered pairs of vertices $ u $ and $ v $ of $ G $. In this paper, we show that the minimum distance–unbalancedness of the merged graph $ C_3\cdot T $ is $ (n+2)(n-3) $, where $ C_3 \cdot T $ is obtained by attaching a tree $ T $ to the cycle $ C_3 $.
- distance–unbalancedness,
- distance–balanced graph,
- Mostar index
Citation: Zhenhua Su, Zikai Tang. Minimum distance–unbalancedness of the merged graph of $ C_3 $ and a tree[J]. AIMS Mathematics, 2024, 9(7): 16863-16875. doi: 10.3934/math.2024818

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Abstract

For a graph $ G $, let $ n_G(u, v) $ be the number of vertices of $ G $ that are strictly closer to $ u $ than to $ v $. The distance–unbalancedness index $ {\rm uB}(G) $ is defined as the sum of $ |n_G(u, v)-n_G(v, u)| $ over all unordered pairs of vertices $ u $ and $ v $ of $ G $. In this paper, we show that the minimum distance–unbalancedness of the merged graph $ C_3\cdot T $ is $ (n+2)(n-3) $, where $ C_3 \cdot T $ is obtained by attaching a tree $ T $ to the cycle $ C_3 $.

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